Geometric Sequences

Front

What is a geometric sequence?

Back

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Front

How do you find the common ratio in a geometric sequence?

Back

The common ratio (r) can be found by dividing any term in the sequence by the previous term, i.e., r = a_n / a_(n-1).

Front

What does a_1 represent in a geometric sequence?

Back

a_1 represents the first term of the geometric sequence.

Front

If the first term of a geometric sequence is 3 and the common ratio is 2, what are the first four terms?

Back

The first four terms are 3, 6, 12, 24.

Front

What is the formula for the nth term of a geometric sequence?

Back

The nth term of a geometric sequence can be calculated using the formula a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.

Front

Find the next three terms of the sequence: 5, 15, 45, ...

Back

The next three terms are 135, 405, 1215.

Front

What is the common ratio for the sequence: 2, 10, 50, 250, ...?

Back

The common ratio is 5.

Front

How do you determine the first term of a geometric sequence if you know the common ratio and the second term?

Back

If you know the common ratio (r) and the second term (a_2), you can find the first term (a_1) using the formula a_1 = a_2 / r.

Front

What is the sum of the first n terms of a geometric sequence?

Back

The sum of the first n terms (S_n) of a geometric sequence can be calculated using the formula S_n = a_1 * (1 - r^n) / (1 - r) for r ≠ 1.

Front

If a geometric sequence has a common ratio of 1, what type of sequence is it?

Back

If the common ratio is 1, the sequence is a constant sequence, where all terms are the same.